Question

3. Two sides AB and BC and median AM of one triangle
ABC are respectively equal to sides PQ and QR and
median PN of ∆ PQR (see Fig.). Show that:







(i) ∆ ABM =∆ PQN
(ii) ∆ ABC = ∆ PQR​

Answer 1

Answer:

Triangle ABM is congruent to triangle PQN

Triangle ABC is congruent to triangle PQR

Answer 2

Answer:

Step-by-step explanation:

Given

AB = PQ

BC = QR

AM = PN

AM and PN are the medians of ΔABC and ΔPQR

     BM = CM = half of BC

     QN = RN = half of QR

To Prove

(i) ΔABM ≅ ΔPQN

(ii) ΔABC ≅ ΔPQR

Proof

(i) Consider ΔABM and ΔPQN

               (Side) AB = PQ {Given}

               (Side) AM = PN {Given}

BC = QR {Given}

Taking half on both sides

1/2 BC = 1/2 QR

(Side) BM = QN {1/2BC = BM and 1/2QR = QN}

∴ ΔABM ≅ ΔPQN [ by SSS congruence rule]

(ii) Consider ΔABC and ΔPQR

     (Side) AB = PQ {Given}

     (Side) BC = QR {Given}

    ∠ABM = ∠PQN {by CPCT ∵ΔABM ≅ ΔPQN}

     ∴  (Angle) ∠ABC = ∠PQR

    ∴ ΔABC ≅ ΔPQR [by SAS congruence rule]

   Hence Proved :)